A sample of 81 calculus students at a large college had a mean mathematics ACT score of 26 with a standard deviation of 6. Find a 95% confidence interval for the mean mathematics ACT score for all calculus students at this college.

Answer: 24.67 ≤ μ ≤ 27.33

Step-by-step explanation:

From the information given: n = 81 students, Sample mean (x bar) = 26 and sample standard deviation (s) = 6.

Since the population standard deviation in unknown we use the student t distribution to estimate the mean. The mean is estimated as follows:

X bar - t (∝/2, n-1) x s/sqrt n ≤ μ ≤ X bar + t (∝/2, n-1) x s/sqrt n

Degrees of freedom, df = n - 1 = 81 - 1 = 80

∝ = 100 - 95 = 5% = 0.05 thus ∝/2 = 0.025

From the t distribution t (0.025,80) = 1.990

Substituting into the formula:

26 - 1.990 x 6/sqrt 81 ≤ μ ≤ 26 + 1.990 x 6/sqrt 81

24.67 ≤ μ ≤ 27.33